cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A237825 Number of partitions of n such that 3*(least part) = greatest part.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 5, 5, 8, 9, 13, 14, 18, 20, 27, 28, 35, 38, 49, 51, 61, 66, 81, 86, 102, 109, 130, 136, 161, 172, 202, 214, 245, 264, 305, 323, 369, 395, 452, 480, 544, 580, 657, 703, 786, 842, 947, 1008, 1124, 1205, 1340, 1432, 1589, 1702, 1886, 2014
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2014

Keywords

Examples

			a(7) = 3 counts these partitions:  331, 3211, 31111.

Links

Crossrefs

Cf. A000041, A117086, A237757, A237828, A237829.
Cf. A118096, A237826, A237827.

Programs

  • Mathematica
    z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
Table[Count[IntegerPartitions[n],?(3#[[-1]]==#[[1]]&)],{n,60}] (* _Harvey P. Dale, May 14 2023 *)
kmax = 57;
Sum[x^(4 k)/Product[1 - x^j, {j, k, 3 k}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, 3*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023

Formula

G.f.: Sum_{k>=1} x^(4*k)/Product_{j=k..3*k} (1-x^j). - Seiichi Manyama, May 14 2023
a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 0.23036554... - Vaclav Kotesovec, Jun 14 2025

A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

Original entry on oeis.org

2, 0, 4, 5, 3, 9, 0, 6, 9, 1, 8, 5, 2, 0, 5, 0, 6, 3, 9, 8, 9, 3, 7, 0, 4, 2, 4, 4, 3, 4, 2, 6, 0, 1, 2, 5, 2, 2, 6, 5, 9, 4, 8, 7, 9, 3, 4, 6, 7, 8, 3, 3, 1, 8, 7, 9, 9, 4, 6, 6, 2, 8, 7, 0, 9, 3, 4, 4, 5, 5, 6, 1, 7, 3, 3, 7, 1, 1, 0, 7, 1, 3, 9, 6, 9, 8, 9, 2, 2, 1, 6, 4, 8, 1, 4, 2, 5, 3, 9, 5, 2, 5, 2, 8, 0, 9
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 01 2024

Keywords

Examples

			2.045390691852050639893704244342601252265948793467833187994662870934455617...

Crossrefs

Cf. A333198, A376621, A376630, A376631, A376658, A376659, A376815.
Cf. A088559.

Programs

  • Mathematica
    RealDigits[E^Sqrt[3*Log[r]^2/4 + 2*PolyLog[2, r^(1/2)] - Pi^2/6] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A376630(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376631(n)^(1/sqrt(n)).
Equals A376815^(1/2). - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2/4 + 2*polylog(2, r^(1/2)) - Pi^2/6)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A376812 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j)^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 5, 5, 5, 7, 8, 10, 13, 14, 16, 19, 21, 25, 29, 33, 40, 45, 50, 57, 64, 72, 81, 93, 104, 117, 134, 148, 165, 185, 204, 227, 253, 280, 310, 345, 381, 422, 469, 514, 567, 625, 685, 753, 825, 903, 990, 1086, 1186, 1297, 1419, 1548, 1692, 1845, 2007
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 05 2024

Keywords

Links

Crossrefs

Cf. A053261, A376626, A376627, A376813, A216222.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1)/2)*Product[1+x^k, {k, 1, n}]^2, {n, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (1 + x^j)^2 * x^j.
a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 1/(4*sqrt(3/2 - 2*sinh(arcsinh(3^(3/2)/2)/3)/sqrt(3))) = 0.27647151570071656262813536...

A363075 Number of partitions of n such that 3*(least part) + 1 = greatest part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 6, 6, 10, 12, 18, 20, 27, 32, 42, 47, 59, 67, 85, 94, 113, 126, 152, 169, 198, 220, 257, 282, 326, 359, 413, 452, 512, 563, 639, 695, 781, 853, 958, 1041, 1161, 1261, 1402, 1524, 1685, 1827, 2021, 2186, 2407, 2604, 2861, 3088, 3385, 3657, 4002, 4316, 4704, 5069, 5531
Offset: 1

Views

Author

Seiichi Manyama, May 17 2023

Keywords

Links

Crossrefs

Cf. A049820, A237828, A363076, A363077.
Cf. A237825.

Programs

  • Mathematica
    nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(3*k - 2))*(1 - x^(3*k - 1))*(1 - x^(3*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(4*k + 1)/(1 - x^k)/(1 - x^(3*k + 1))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 19 2025 *)
  • PARI
    my(N=70, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(4*k+1)/prod(j=k, 3*k+1, 1-x^j))))

Formula

G.f.: Sum_{k>=1} x^(4*k+1)/Product_{j=k..3*k+1} (1-x^j).
a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 0.33761... - Vaclav Kotesovec, Jun 20 2025
Showing 1-4 of 4 results.

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